\(\int (a+b \log (c x^n))^2 \log (1+e x) \, dx\) [19]

Optimal result

Integrand size = 19, antiderivative size = 193 \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=2 a b n x-6 b^2 n^2 x+2 b^2 n x \log \left (c x^n\right )+2 b n x \left (a+b \log \left (c x^n\right )\right )-x \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 n^2 (1+e x) \log (1+e x)}{e}-\frac {2 b n (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}-\frac {2 b^2 n^2 \operatorname {PolyLog}(2,-e x)}{e}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,-e x)}{e}-\frac {2 b^2 n^2 \operatorname {PolyLog}(3,-e x)}{e} \] Output:

2*a*b*n*x-6*b^2*n^2*x+2*b^2*n*x*ln(c*x^n)+2*b*n*x*(a+b*ln(c*x^n))-x*(a+b*l 
n(c*x^n))^2+2*b^2*n^2*(e*x+1)*ln(e*x+1)/e-2*b*n*(e*x+1)*(a+b*ln(c*x^n))*ln 
(e*x+1)/e+(e*x+1)*(a+b*ln(c*x^n))^2*ln(e*x+1)/e-2*b^2*n^2*polylog(2,-e*x)/ 
e+2*b*n*(a+b*ln(c*x^n))*polylog(2,-e*x)/e-2*b^2*n^2*polylog(3,-e*x)/e
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.52 \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\frac {-a^2 e x+4 a b e n x-6 b^2 e n^2 x-2 a b e x \log \left (c x^n\right )+4 b^2 e n x \log \left (c x^n\right )-b^2 e x \log ^2\left (c x^n\right )+a^2 \log (1+e x)-2 a b n \log (1+e x)+2 b^2 n^2 \log (1+e x)+a^2 e x \log (1+e x)-2 a b e n x \log (1+e x)+2 b^2 e n^2 x \log (1+e x)+2 a b \log \left (c x^n\right ) \log (1+e x)-2 b^2 n \log \left (c x^n\right ) \log (1+e x)+2 a b e x \log \left (c x^n\right ) \log (1+e x)-2 b^2 e n x \log \left (c x^n\right ) \log (1+e x)+b^2 \log ^2\left (c x^n\right ) \log (1+e x)+b^2 e x \log ^2\left (c x^n\right ) \log (1+e x)+2 b n \left (a-b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,-e x)-2 b^2 n^2 \operatorname {PolyLog}(3,-e x)}{e} \] Input:

Integrate[(a + b*Log[c*x^n])^2*Log[1 + e*x],x]
 

Output:

(-(a^2*e*x) + 4*a*b*e*n*x - 6*b^2*e*n^2*x - 2*a*b*e*x*Log[c*x^n] + 4*b^2*e 
*n*x*Log[c*x^n] - b^2*e*x*Log[c*x^n]^2 + a^2*Log[1 + e*x] - 2*a*b*n*Log[1 
+ e*x] + 2*b^2*n^2*Log[1 + e*x] + a^2*e*x*Log[1 + e*x] - 2*a*b*e*n*x*Log[1 
 + e*x] + 2*b^2*e*n^2*x*Log[1 + e*x] + 2*a*b*Log[c*x^n]*Log[1 + e*x] - 2*b 
^2*n*Log[c*x^n]*Log[1 + e*x] + 2*a*b*e*x*Log[c*x^n]*Log[1 + e*x] - 2*b^2*e 
*n*x*Log[c*x^n]*Log[1 + e*x] + b^2*Log[c*x^n]^2*Log[1 + e*x] + b^2*e*x*Log 
[c*x^n]^2*Log[1 + e*x] + 2*b*n*(a - b*n + b*Log[c*x^n])*PolyLog[2, -(e*x)] 
 - 2*b^2*n^2*PolyLog[3, -(e*x)])/e
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2817, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2 \, dx\)

\(\Big \downarrow \) 2817

\(\displaystyle -2 b n \int \left (-a-b \log \left (c x^n\right )+\frac {(e x+1) \left (a+b \log \left (c x^n\right )\right ) \log (e x+1)}{e x}\right )dx+\frac {(e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{e}-x \left (a+b \log \left (c x^n\right )\right )^2\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 b n \left (-\frac {\operatorname {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {(e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{e}-x \left (a+b \log \left (c x^n\right )\right )-a x-b x \log \left (c x^n\right )+\frac {b n \operatorname {PolyLog}(2,-e x)}{e}+\frac {b n \operatorname {PolyLog}(3,-e x)}{e}-\frac {b n (e x+1) \log (e x+1)}{e}+3 b n x\right )+\frac {(e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{e}-x \left (a+b \log \left (c x^n\right )\right )^2\)

Input:

Int[(a + b*Log[c*x^n])^2*Log[1 + e*x],x]
 

Output:

-(x*(a + b*Log[c*x^n])^2) + ((1 + e*x)*(a + b*Log[c*x^n])^2*Log[1 + e*x])/ 
e - 2*b*n*(-(a*x) + 3*b*n*x - b*x*Log[c*x^n] - x*(a + b*Log[c*x^n]) - (b*n 
*(1 + e*x)*Log[1 + e*x])/e + ((1 + e*x)*(a + b*Log[c*x^n])*Log[1 + e*x])/e 
 + (b*n*PolyLog[2, -(e*x)])/e - ((a + b*Log[c*x^n])*PolyLog[2, -(e*x)])/e 
+ (b*n*PolyLog[3, -(e*x)])/e)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2817

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[Log[d*(e + f*x^m)^r], x]}, 
Simp[(a + b*Log[c*x^n])^p   u, x] - Simp[b*n*p   Int[(a + b*Log[c*x^n])^(p 
- 1)/x   u, x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] 
&& RationalQ[m] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 
 1] && EqQ[m, 1] && EqQ[d*e, 1]))
 

Maple [F]

\[\int {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (e x +1\right )d x\]

Input:

int((a+b*ln(c*x^n))^2*ln(e*x+1),x)
 

Output:

int((a+b*ln(c*x^n))^2*ln(e*x+1),x)
 

Fricas [F]

\[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left (e x + 1\right ) \,d x } \] Input:

integrate((a+b*log(c*x^n))^2*log(e*x+1),x, algorithm="fricas")
 

Output:

integral(b^2*log(c*x^n)^2*log(e*x + 1) + 2*a*b*log(c*x^n)*log(e*x + 1) + a 
^2*log(e*x + 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*x**n))**2*ln(e*x+1),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left (e x + 1\right ) \,d x } \] Input:

integrate((a+b*log(c*x^n))^2*log(e*x+1),x, algorithm="maxima")
 

Output:

-(b^2*e*x - (b^2*e*x + b^2)*log(e*x + 1))*log(x^n)^2/e + integrate(((b^2*e 
*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x*log(e*x + 1) + 2*(b^2*e*n*x - (b^2*n 
 + ((e*n - e*log(c))*b^2 - a*b*e)*x)*log(e*x + 1))*log(x^n))/x, x)/e
 

Giac [F]

\[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left (e x + 1\right ) \,d x } \] Input:

integrate((a+b*log(c*x^n))^2*log(e*x+1),x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^2*log(e*x + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\int \ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \] Input:

int(log(e*x + 1)*(a + b*log(c*x^n))^2,x)
 

Output:

int(log(e*x + 1)*(a + b*log(c*x^n))^2, x)
 

Reduce [F]

\[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\frac {-3 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e \,x^{2}+x}d x \right ) b^{2} n -6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{2}+x}d x \right ) a b n +6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{2}+x}d x \right ) b^{2} n^{2}+3 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e n x +6 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right ) a b e n x -6 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right ) b^{2} e \,n^{2} x +3 \,\mathrm {log}\left (e x +1\right ) a^{2} e n x +3 \,\mathrm {log}\left (e x +1\right ) a^{2} n -6 \,\mathrm {log}\left (e x +1\right ) a b e \,n^{2} x -6 \,\mathrm {log}\left (e x +1\right ) a b \,n^{2}+6 \,\mathrm {log}\left (e x +1\right ) b^{2} e \,n^{3} x +6 \,\mathrm {log}\left (e x +1\right ) b^{2} n^{3}+\mathrm {log}\left (x^{n} c \right )^{3} b^{2}+3 \mathrm {log}\left (x^{n} c \right )^{2} a b -3 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e n x -3 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} n -6 \,\mathrm {log}\left (x^{n} c \right ) a b e n x +12 \,\mathrm {log}\left (x^{n} c \right ) b^{2} e \,n^{2} x -3 a^{2} e n x +12 a b e \,n^{2} x -18 b^{2} e \,n^{3} x}{3 e n} \] Input:

int((a+b*log(c*x^n))^2*log(e*x+1),x)
 

Output:

( - 3*int(log(x**n*c)**2/(e*x**2 + x),x)*b**2*n - 6*int(log(x**n*c)/(e*x** 
2 + x),x)*a*b*n + 6*int(log(x**n*c)/(e*x**2 + x),x)*b**2*n**2 + 3*log(e*x 
+ 1)*log(x**n*c)**2*b**2*e*n*x + 6*log(e*x + 1)*log(x**n*c)*a*b*e*n*x - 6* 
log(e*x + 1)*log(x**n*c)*b**2*e*n**2*x + 3*log(e*x + 1)*a**2*e*n*x + 3*log 
(e*x + 1)*a**2*n - 6*log(e*x + 1)*a*b*e*n**2*x - 6*log(e*x + 1)*a*b*n**2 + 
 6*log(e*x + 1)*b**2*e*n**3*x + 6*log(e*x + 1)*b**2*n**3 + log(x**n*c)**3* 
b**2 + 3*log(x**n*c)**2*a*b - 3*log(x**n*c)**2*b**2*e*n*x - 3*log(x**n*c)* 
*2*b**2*n - 6*log(x**n*c)*a*b*e*n*x + 12*log(x**n*c)*b**2*e*n**2*x - 3*a** 
2*e*n*x + 12*a*b*e*n**2*x - 18*b**2*e*n**3*x)/(3*e*n)